Random fuzzy renewal process

This paper attempts to discuss a random fuzzy renewal process based on random fuzzy theory. The interarrival times are characterized as nonnegative random fuzzy variables which is a more reasonable consideration in the real world. Under this setting, the rate of the random fuzzy renewal process is discussed and a random fuzzy elementary renewal theorem is presented. Furthermore, the expected value of renewals in an arbitrary interval is investigated and Blackwell’s theorem in random fuzzy sense is also established.     

Delayed renewal process with uncertain interarrival times

Delayed renewal process is a special type of renewal process in which the first interarrival time is quite different from the others. This paper first proposes an uncertain delayed renewal process whose interarrival times are regarded as uncertain variables. Then it gives an uncertainty distribution of delayed renewal process and an elementary delayed renewal theorem.     

Uniform convergence of fuzzy random renewal process

Recently, Zhao et al. (in Fuzzy Optimization and Decision Making 2007 6, 279–295) presented a fuzzy random elementary renewal theorem and fuzzy random renewal reward theorem in the fuzzy random process. In this paper, we study the convergence of fuzzy random renewal variable and of the total rewards earned by time t with respect to the extended Hausdorff metrics d _∞ and d ₁. Using this convergence information and applying the uniform convergence theorem, we provide some new versions of the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem.     

Poisson process with fuzzy rates

In a stochastic homogeneous Poisson process, interarrival times are independent and identically distributed (iid) exponential random variables whose parameter is called the rate of the process. By using fuzzy variables to describe the parameter, a Poisson process whose rates are fuzzy variables is established. Based on the random fuzzy theory, relationship between the renewal number and fuzzy rates is discussed. As an application, a random fuzzy compound Poisson process is investigated.     

A CLT for renewal processes with a finite set of interarrival distributions

We prove a central limit theorem for a renewal process based on a sequence of independent non-negative interarrival times whose distributions are taken from a finite set. The result extends the classical central limit theorem obtained by Takács (1956).     

Fractional Liu process with application to finance

As a fuzzy counterpart of Brownian motion, Liu process has attracted more and more attention in the recent literature. In this paper, the concept of fractional Liu process is proposed as an extension of Liu process. Furthermore, we obtain the expressions of the membership functions, expected values and variances of arithmetic and geometric fractional Liu processes for each fixed time. As an application, geometric fractional Liu process is assumed to characterize the stock price, which formulates a new fuzzy stock model. Based on this proposed model, European option pricing formulas are gained and two numerical examples are given with different parameters.     

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.     

On performance comparison of MR/GI/1 queues

In this paper we are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. We start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrivai times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. [14].Then, in order to keep the marginal distributions of the interarrivai times constant, we build a particular transition matrix for the underlying Markov chain depending on a single parameter,p. This Markov renewal process is used in the Patuwo et al. [14] problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only onp. As constructed, the interarrival time distributions do not depend onp so that the effects we find depend only on correlation in the arrival process.As a result of this latter construction, we find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of our results are clear.     

Interval optimization based line sampling method for fuzzy and random reliability analysis

For structural system with fuzzy variables as well as random variables, a novel algorithm for obtaining membership function of fuzzy reliability is presented on interval optimization based Line Sampling (LS) method. In the presented algorithm, the value domain of the fuzzy variables under the given membership level is firstly obtained according to their membership functions. Then, in the value domain of the fuzzy variables, bounds of reliability of the structure are obtained by the nesting analysis of the interval optimization, which is performed by modern heuristic methods, and reliability analysis, which is achieved by the LS method in the reduced space of the random variables. In this way the uncertainties of the input variables are propagated to the safety measurement of the structure, and the membership function of the fuzzy reliability is obtained. The presented algorithm not only inherits the advantage of the direct Monte Carlo method in propagating and distinguishing the fuzzy and random uncertainties, but also can improve the computational efficiency tremendously in case of acceptable precision. Several examples are used to illustrate the advantages of the presented algorithm.     

Uniform estimates for the finite-time ruin probability in the dependent renewal risk model

This paper investigates the finite-time ruin probability in the dependent renewal risk model, where the claim sizes are independent and identically distributed random variables with strongly subexponential tails, and the interarrival times are negatively dependent. We establish an asymptotic estimate, which holds uniformly for the time horizon varying in the positive half line.     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

A curious renewal process average

The lifetimes of a renewal process observed during a fixed interval (0, t] are smaller, on the average, than the process mean lifetime; it is shown that the mean observed lifetime has a particularly simple form.     

On a transient renewal function of dominated variation

We show that under a certain condition the renewal function of a transient renewal process varies dominatedly if and only if the interarrival times distribution varies dominatedly. This way the result of [1] is generalized.     

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

The two-moment three-parameter decomposition approximation of queueing networks with exponential residual renewal processes

We propose a two-moment three-parameter decomposition approximation of general open queueing networks by which both autocorrelation and cross correlation are accounted for. Each arrival process is approximated as an exponential residual (ER) renewal process that is characterized by three parameters: intensity, residue, and decrement. While the ER renewal process is adopted for modeling autocorrelated processes, the innovations method is used for modeling the cross correlation between randomly split streams. As the interarrival times of an ER renewal process follow a two-stage mixed generalized Erlang distribution, viz., MGE(2), each station is analyzed as an MGE(2)/G/1 system for the approximate mean waiting time. Variability functions are also used in network equations for a more accurate modeling of the propagation of cross correlations in queueing networks. Since an ER renewal process is a special case of a Markovian arrival process (MAP), the value of the variability function is determined by a MAP/MAP/1 approximation of the departure process. Numerical results show that our proposed approach greatly improves the performance of the parametric decomposition approximation of open queueing networks.     

A modified insurance risk process with uncertainty

An insurance risk process is traditionally considered by describing the claim process via a renewal reward process and assuming the total premium to be proportional to the time with a constant ratio. It is usually modeled as a stochastic process such as the compound Poisson process, and historical data are collected and employed to estimate the corresponding parameters of probability distributions. However, there exists the case of lack of data such as for a new insurance product. An alternative way is to estimate the parameters based on experts’ subjective belief and information. Therefore, it is necessary to employ the uncertain process to model the insurance risk process. In this paper, we propose a modified insurance risk process in which both the claim process and the premium process are assumed to be renewal reward processes with uncertain factors. Then we give the inverse uncertainty distribution of the modified process at each time. On this basis, we derive the ruin index which has an explicit expression based on given uncertainty distributions.     

A deteriorating system with its repairman having multiple vacations

This paper considers a repairable system with a repairman, who can take multiple vacations. If the system fails and the repairman is on vacation, it will wait for repair until the repairman is available. Assume that the system cannot be repaired “as good as new” after failures. Under these assumptions, using the geometric process and the supplementary variable technique, some important reliability indexes are derived, such as the system reliability, availability, rate of occurrence of failures, etc. According to the renewal reward theorem, the explicit expression of the expected profit per unit time is obtained. Finally, a numerical example is given to illustrate that there exists an optimal replacement policy N∗, which maximizes the value of the expected profit rate after a long time run.     

Stationary distributions of semistochastic processes with disturbances at random times and with random severity

We consider a semistochastic continuous-time continuous-state space random process that undergoes downward disturbances with random severity occurring at random times. Between two consecutive disturbances, the evolution is deterministic, given by an autonomous ordinary differential equation. The times of occurrence of the disturbances are distributed according to a general renewal process. At each disturbance, the process gets multiplied by a continuous random variable (“severity”) supported on [0,1). The inter-disturbance time intervals and the severities are assumed to be independent random variables that also do not depend on the history.We derive an explicit expression for the conditional density connecting two consecutive post-disturbance levels, and an integral equation for the stationary distribution of the post-disturbance levels. We obtain an explicit expression for the stationary distribution of the random process. Several concrete examples are considered to illustrate the methods for solving the integral equations that occur.     

Counting the number of renewals during a random interval in a discrete-time delayed renewal process

In a discrete-time delayed renewal process, we study the distribution of the number of renewals during a random interval. We obtain closed-form expressions for the probability mass function and binomial moments of this number for various distributions of the random interval and interrenewal times.     

On stationary renewal reward processes where most rewards are zero

We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings. Received: 1 March 1999 / Revised version: 2 August 1999 /?Published online: 31 May 2000